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Variational Kernel Design for Internal Noise: Gaussian Chaos Noise, Representation Compatibility, and Reliable Deep Learning

Ziran Liu

Abstract

Internal noise in deep networks is usually inherited from heuristics such as dropout, hard masking, or additive perturbation. We ask two questions: what correlation geometry should internal noise have, and is the implemented perturbation compatible with the representations it acts on? We answer these questions through Variational Kernel Design (VKD), a framework in which a noise mechanism is specified by a law family, a correlation kernel, and an injection operator, and is derived from learning desiderata. In a solved spatial subfamily, a quadratic maximum-entropy principle over latent log-fields yields a Gaussian optimizer with precision given by the Dirichlet Laplacian, so the induced geometry is the Dirichlet Green kernel. Wick normalization then gives a canonical positive mean-one gate, Gaussian Chaos Noise (GCh). For the sample-wise gate used in practice, we prove exact Gaussian control of pairwise log-ratio deformation, margin-sensitive ranking stability, and an exact expected intrinsic roughness budget; hard binary masks instead induce singular or coherence-amplified distortions on positive coherent representations. On ImageNet and ImageNet-C, GCh consistently improves calibration and under shift also improves NLL at competitive accuracy.

Variational Kernel Design for Internal Noise: Gaussian Chaos Noise, Representation Compatibility, and Reliable Deep Learning

Abstract

Internal noise in deep networks is usually inherited from heuristics such as dropout, hard masking, or additive perturbation. We ask two questions: what correlation geometry should internal noise have, and is the implemented perturbation compatible with the representations it acts on? We answer these questions through Variational Kernel Design (VKD), a framework in which a noise mechanism is specified by a law family, a correlation kernel, and an injection operator, and is derived from learning desiderata. In a solved spatial subfamily, a quadratic maximum-entropy principle over latent log-fields yields a Gaussian optimizer with precision given by the Dirichlet Laplacian, so the induced geometry is the Dirichlet Green kernel. Wick normalization then gives a canonical positive mean-one gate, Gaussian Chaos Noise (GCh). For the sample-wise gate used in practice, we prove exact Gaussian control of pairwise log-ratio deformation, margin-sensitive ranking stability, and an exact expected intrinsic roughness budget; hard binary masks instead induce singular or coherence-amplified distortions on positive coherent representations. On ImageNet and ImageNet-C, GCh consistently improves calibration and under shift also improves NLL at competitive accuracy.
Paper Structure (111 sections, 22 theorems, 144 equations, 1 figure, 12 tables, 1 algorithm)

This paper contains 111 sections, 22 theorems, 144 equations, 1 figure, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scope of the theoretical claims.
  3. Contributions.
  4. A practical way to read the paper.
  5. Roadmap.
  6. Paper in one sentence.
  7. What is classical and what is new.
  8. Related Work
  9. Noise injection and regularization in deep networks.
  10. Calibration and reliability under shift.
  11. Robustness to corruptions and distribution shift.
  12. Variational Kernel Design as a Compositional Design System
  13. Mechanism space: VKD as a design system
  14. From desiderata to admissible mechanism classes
  15. A two-layer view: mechanism and compatibility
  16. ...and 96 more sections

Key Result

Theorem 5.1

Let $Q\succ 0$ be symmetric positive definite on $\mathbb{R}^U$, let $n=|U|$, and let $\varepsilon>0$. Then the variational problem eq:master_variational_problem has a unique optimizer Equivalently, with precision matrix Moreover, for every $p\in\mathcal{A}(Q,\varepsilon)$, so the optimizer is unique. Its entropy is

Figures (1)

  • Figure 1: VKD as a compositional design system. The top row shows the general design logic: learning desiderata define an admissible mechanism class, from which a mechanism and realization map are derived and then evaluated through compatibility observables on a target representation regime. The bottom row shows the solved instance of this paper. A quadratic operator budget with $Q=L_U$ yields a Gaussian log-field with covariance proportional to the Dirichlet Green kernel. This latent field admits a canonical exact realization through Wick normalization and an optimization-friendly implemented realization through sample-wise mean-one normalization. The resulting deployed gate is analyzed through pairwise, ranking, roughness, and topological observables, and then tested empirically through calibration and reliability metrics.

Theorems & Definitions (45)

  • Definition 3.1: VKD mechanism
  • Remark 3.2: VKD is compositional, not temporal
  • Theorem 5.1: Design theorem for the quadratic VKD subfamily
  • proof : Proof sketch
  • Corollary 5.2: Operator-forced geometry in the Dirichlet instantiation
  • Remark 5.3: What is and is not "forced"
  • Theorem 5.4: Canonical realization in the solved VKD subfamily
  • proof
  • Proposition 5.5: Effective one-parameter scaling
  • proof
  • ...and 35 more